Chapter 4 - Vector algebra

Question 1

What are direction cosines?

Answer 1

The direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three coordinate axes.

Question 2

Define OP’s direction cosines (l, m, n) in regards to x, y and z.

Answer 2

l = x / r

m = y / r

n = z / r

Question 3

If P har coordinates (2, -1, 3), find the length oP and the direction cosines of OP.

Answer 3

OP² = (2)² + (-1)² + (3)²

= 4 + 1 + 9

This gives OP = sqrt(14)

Direction cosines are:

l = 2*sqrt(1/14)

m = -sqrt(1/14)

n = 3sqrt(1/14)

Question 4

When are two vectors equal?

Answer 4

Two vectors a and b are equal if and only if they have the same modulus and the same direction and sense.

In component form, two vectors are equal if and only if the components are equal, that is:

a₁ = b₁, a₂ = b₂, a₃ = b₃

Question 5

If k is a scalar and the vectors are related by a = kb then:

if k > 0, a is a vector in the ____ as b with magnitude k times the magnitude of b.

Answer 5

same direction

Question 6

If k is a scalar and the vectors are related by a = kb then:

if k < 0, a is a vector in the ____ as b with magnitude k times the magnitude of b.

Answer 6

opposite direction

Question 7

What is a vectors modulus?

Answer 7

A vectors modulus is a vectors length or magnitude. It’s written as |a| or |OA| (with arrow on top).

Question 8

What is a unit vector?

Answer 8

A unit vector is a vector with modulus 1.

It’s written with a hat: â

â = a / |a|

Question 9

What does the triangle law say about vector addition?

Answer 9

If two vectors a and b are represented in magnitude and direction by the two sides of a triangle taken in order then their sum is represented in magnitude and direction by the closing third side.

Question 10

What does the commutative law say?

Answer 10

a + b = b + a

Order does not matter

Question 11

What does the associative law say?

Answer 11

The brackets do not matter and can be omitted.

(a+b) + c = a + (b + c)

Question 12

What does the distributive law say?

Answer 12

k(a + b) = ka + kb

We can multiply brackets out by the usual laws of algebra.

Question 13

How do you subtract vectors?

Answer 13

We define subtraction in the obvious way:

a - b = a + (-b)

Where -b is the same vector as b, but facing the opposite way.

Question 14

What letters are most commonly used to describe the three coordinate unit vectors in 3D-space?

Answer 14

i, j and k.

i for x-axis, j for y-axis, and k for z-axis.

Question 15

How do you add the vectors (a₁, a₂, a₃) and b₁, b₂, b₃)?

Answer 15

By adding each of the respective components together:

(a₁+b₁, a₂+b₂, a₃+b₃)

Question 16

How do you multiply a vector with a scalar? (Component form)

Answer 16

If a = (a₁, a₂, a₃), then ka is:

(ka₁, ka₂, ka₃)

Question 17

How is the scalar (or dot or inner) product of two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) defined? (Both component and geometrical form)

Answer 17

In components:

a • b = a₁b₁ + a₂b₂ + a₃b₃ (= a scalar number)

a • b = | a | | b | cos ß

Question 18

Consider that the scalar product of two vectors is equal to zero. What does this imply?

Answer 18

The two vectors are perpendicular.

Question 19

a • b = 0
can mean three things. What are they?

Answer 19

either a = 0,

or b = 0,

or the angle between a and b is 90 degrees (they are perpendicular).

Question 20

Does a • b = a • c
mean that b = c?

Answer 20

Not necessarily. There are three possible solutions:

Either, b = c

or a = 0

or a is perpendicular to b - c.

Question 21

Consider the unit vectors i, j and k.

What is i • j
equal to?

Answer 21

i • j = j • k = k • i = 0

because the unit vectors are mutually perpendicular.

Question 22

Given the two vectors a and b, what is the geometrical definition of the vector product?

Answer 22

a x b = | a | | b | sin ß ñ,

where ß is the angle between a and b (and between 0 and π), and ñ is the unit vector perpendicular to both a and b, such that a, b, ñ form a right-handed set.

The vector product of two vectors is itself a vector.

Question 23

What does the anti-commutative law for vector products say?

Answer 23

a x b = -(b x a)

Question 24

The distributive law over addition for vector products states that

(a x b) + (a x c)

can be written as ___

Answer 24

a x (b + c)

Question 25

Given the two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), how can the vector product be defined in component form?

Answer 25

a x b can be defined in determinant form (actually an accepted misuse of the determinant form)

Question 26

The vector product of two vectors is also the area of ____

Answer 26

The parallelogram defined by the two vectors (++ (du forstår!))

Question 27

Where the vector product defines the area of the parallelogram, the triple scalar product defines the _____

Answer 27

Area of the parallelepiped

Question 28

In cartesian/component form, the triple scalar product can be written as:

Answer 28

The determinant

Question 29

Describe in a few steps how to find the area of a triangle using vector product, given the three vertices A, B and C of the triangle.

Answer 29

1. Find two vectors starting from the same point, e.g. AB and AC.
2. Find the vector product (AB x AC).
3. Since the area of the parallelogram is the same as the length of the vector product, the area of the triangle is half of that.

Area of triangle = 1/2 * |(AB x AC)|

Question 30

Describe in a few steps how to find the constant k such that the three vectors (7, 3, -1), (1, -1, 7) and (2, -4, k) are coplanar.

(You don’t need to do the calculations)

Answer 30

1. If the three vectors are coplanar, they have a common perpendicular vector. Since we have two whole vectors, we can find the vector product of those. The vector product is itself a vector, and is perpendicular to the two vectors.
2. We now have the perpendicular vector (a). We want a • (2, -4, k) = 0
   to be true. Therefore, we manipulate the scalar product to get k alone.

The answer to this particular problem is 24.

Question 31

Describe in a few steps how to solve this problem:

If a and b are perpendicular, simplify (a - 3b) • (8a + 5b)

Answer 31

1. When multiplying for scalar product like this, you can solve the expression like a regular algebraic expression. This particular problem will result in (8a²-19ab-15b²).
2. Since we know that a and b are perpendicular, (19ab) will result in 0, since ab = 0.
3. Thus, the simplified expression is 8a²-15b².

Question 32

Describe in a few steps how to solve this problem:

Find the angle between p = 8i - 5j and q = -11i + 9j

Answer 32

1. We know that |p| |q| cosß = 8*(-11) + (-5)*9.

The left side is the geometrical definition of the scalar product, while the rigt side is the definition of the scalar product on component form.

2. From there, we manipulate the equation to get cosß alone on one side of the equation.